A damning independent review published last week found Mr Welby – the most senior bishop within the Church of England – and other church officers should have formally reported Smyth in 2013 to police in the UK and authorities in South Africa.
Smyth was accused of attacking dozens of boys, including those he met at Christian camps, in the UK in the 1970s and 1980s.
The barrister and senior member of a Christian charity then moved to Zimbabwe and later South Africa, where he abused up to 100 boys aged 13 to 17, the Makin review added.
By 2013, the Church of England “knew, at the highest level” about Smyth’s abuse, including Mr Welby who took up the Church’s top job that year.
If he and other Church officers had reported this to police in the UK and authorities in South Africa at that time, “John Smyth could have [been] brought to justice at a much earlier point”, the independent report said.
Mr Welby had previously resisted calls to step aside over his response to the case since 2013.
But amid mounting pressure, he said in a statement on Tuesday he must take “personal and institutional responsibility”.
The prime minister’s spokeswoman said Keir Starmer “respects the decision” to step down and his thoughts “first and foremost, remain with all the victims”.
